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Mirrors > Home > MPE Home > Th. List > subrgid | Structured version Unicode version |
Description: Every ring is a subring of itself. (Contributed by Stefan O'Rear, 30-Nov-2014.) |
Ref | Expression |
---|---|
subrgss.1 |
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Ref | Expression |
---|---|
subrgid |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subrgss.1 |
. . . . 5
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2 | 1 | ressid 14344 |
. . . 4
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3 | id 22 |
. . . 4
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4 | 2, 3 | eqeltrd 2539 |
. . 3
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5 | 4 | ancli 551 |
. 2
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6 | eqid 2451 |
. . . 4
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7 | 1, 6 | rngidcl 16780 |
. . 3
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8 | ssid 3476 |
. . 3
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9 | 7, 8 | jctil 537 |
. 2
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10 | 1, 6 | issubrg 16980 |
. 2
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11 | 5, 9, 10 | sylanbrc 664 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1952 ax-ext 2430 ax-sep 4514 ax-nul 4522 ax-pow 4571 ax-pr 4632 ax-un 6475 ax-cnex 9442 ax-resscn 9443 ax-1cn 9444 ax-icn 9445 ax-addcl 9446 ax-addrcl 9447 ax-mulcl 9448 ax-mulrcl 9449 ax-mulcom 9450 ax-addass 9451 ax-mulass 9452 ax-distr 9453 ax-i2m1 9454 ax-1ne0 9455 ax-1rid 9456 ax-rnegex 9457 ax-rrecex 9458 ax-cnre 9459 ax-pre-lttri 9460 ax-pre-lttrn 9461 ax-pre-ltadd 9462 ax-pre-mulgt0 9463 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2264 df-mo 2265 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2601 df-ne 2646 df-nel 2647 df-ral 2800 df-rex 2801 df-reu 2802 df-rmo 2803 df-rab 2804 df-v 3073 df-sbc 3288 df-csb 3390 df-dif 3432 df-un 3434 df-in 3436 df-ss 3443 df-pss 3445 df-nul 3739 df-if 3893 df-pw 3963 df-sn 3979 df-pr 3981 df-tp 3983 df-op 3985 df-uni 4193 df-iun 4274 df-br 4394 df-opab 4452 df-mpt 4453 df-tr 4487 df-eprel 4733 df-id 4737 df-po 4742 df-so 4743 df-fr 4780 df-we 4782 df-ord 4823 df-on 4824 df-lim 4825 df-suc 4826 df-xp 4947 df-rel 4948 df-cnv 4949 df-co 4950 df-dm 4951 df-rn 4952 df-res 4953 df-ima 4954 df-iota 5482 df-fun 5521 df-fn 5522 df-f 5523 df-f1 5524 df-fo 5525 df-f1o 5526 df-fv 5527 df-riota 6154 df-ov 6196 df-oprab 6197 df-mpt2 6198 df-om 6580 df-recs 6935 df-rdg 6969 df-er 7204 df-en 7414 df-dom 7415 df-sdom 7416 df-pnf 9524 df-mnf 9525 df-xr 9526 df-ltxr 9527 df-le 9528 df-sub 9701 df-neg 9702 df-nn 10427 df-2 10484 df-ndx 14288 df-slot 14289 df-base 14290 df-sets 14291 df-ress 14292 df-plusg 14362 df-0g 14491 df-mnd 15526 df-mgp 16706 df-ur 16718 df-rng 16762 df-subrg 16978 |
This theorem is referenced by: subrgmre 17004 rlmlmod 17401 rlmassa 17512 aspval 17514 evlrhm 17727 evlsscasrng 17728 evlsca 17729 evlsvarsrng 17730 evlvar 17731 mpfsubrg 17734 evl1sca 17886 evl1var 17888 evls1scasrng 17891 evls1varsrng 17892 pf1subrg 17900 pf1ind 17907 evl1gsumadd 17910 evl1varpw 17913 rlmnlm 20394 rlmbn 20998 dvply2 21878 dvnply 21880 taylply 21960 mzpmfp 29224 mzpmfpOLD 29225 rgspnval 29666 rgspncl 29667 |
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